THE EFFECT OF MATERIAL AND PROCESS PARAMETERS ON IMPELLER

TORQUE AND POWER CONSUMPTION IN A BLADED MIXER

by

PAVITHRA VALLIAPPAN

A thesis submitted to the

School of Graduate Studies

Rutgers, The State University of New Jersey

In partial fulfillment of the requirements

For the degree of

Master of Science

Graduate Program in Chemical and Biochemical Engineering

Written under the direction of

Benjamin J. Glasser

And approved by

______________________________

______________________________

______________________________

New Brunswick, New Jersey

October, 2018

ii

ABSTRACT OF THE THESIS

The Effect of Material and Process Parameters on Impeller Torque and Power

Consumption in a Bladed Mixer

By PAVITHRA VALLIAPPAN

Thesis Director:

Dr. Benjamin J. Glasser

A large number of industries including catalytic, chemical, cosmetic, food, and pharmaceutical

industries frequently handle powders or granular materials in various unit operations. A

cylindrical mixer mechanically agitated by an impeller blade is a common, industrially relevant

geometry in numerous particle processing technologies. A bladed mixer has a capability of

handling a wide variety of solid and liquid systems: free flowing and cohesive powders, pastes,

or suspensions. The torque needed to move the impeller, provides insight into flow behavior

of the material and can be monitored on the bench scale, pilot scale, and manufacturing scale

so that it could provide useful information for scale-up and process monitoring & control.

Experimental measurements of the agitation torque exerted on a particle bed and the power

draw for the motor driving the impeller blades in a mixing process were conducted to

investigate the impact of particle properties and blade geometry as a function of the blade

rotation rate. It was found that the torque exerted on a granular bed and the power

consumption were a strong function of the impeller blade configuration, the position of the

blades in a deep granular bed, the fill height of the glass beads, and the size and friction

coefficient of the particles. It was observed that the time-averaged torque and power

iii

consumption for different particle sizes qualitatively scaled with particle diameter. A scale-up

relationship for a deep granular bed was developed: the time-averaged torque and average

adjusted power consumption scaled with the square of the material fill height.

iv

ACKNOWLEDGEMENT

Firstly, I would like to express my sincere gratitude to my mentor and advisor Dr. Benjamin

J Glasser for the continuous support of my Masters study, for his patience, motivation,

and immense knowledge. His guidance helped me immensely throughout my research,

professionally and personally. I could not have imagined having a better advisor and

mentor for my Masters study.

I would also like to thank the rest of my thesis committee: Dr. Masanori Hara and Dr.

Ravendra Singh, for their time, availability, insightful comments and suggestions.

With a special mention to Dr. Veerakiet Boonkanokwong without whose constant help

and support I could not have completed this thesis. I also thank my fellow colleagues who

were a part of Dr. Glasser’s Particle Technology Research Group and in the Engineering

Research Center for Structured Organic Particulate Systems (C-SOPS) at Rutgers

University: Rohan Frank and Anusha Noorithaya for all the stimulating discussions, their

friendship and support.

To all my friends at Rutgers University, and my extended family - the DOT group, thank

you for being my family so far away from home.

Finally, I must express my very profound gratitude to my parents and my brother for

providing me with love, unfailing support and continuous encouragement throughout my

years of study and through the process of researching and writing this thesis. This

accomplishment would not have been possible without them. Thank you.

v

TABLE OF CONTENTS

Abstract of Thesis…………………………………………………………………………..…………………………..….ii

Acknowledgement………………………………………………………………………………..……………………….iv

Table of Contents…………………………………………………………………………….……………………….…….v

List of Figures………………………………………………………………………………….…………………………….vii

1. Introduction………………………………………………………………………………….……………………......1

2. Model and Experimental Set Up…………………………………………………………………….…………6

2.1. Experimental Set Up and Mixer Geometry………………………………………………….…….6

2.2. Materials…………………………………………………………………………………………………………..7

2.3. Impeller Torque and Power Measurement………………………………………………………..8

2.4. Particle Surface Roughness.............................................................................…..11

2.5. Figures………………………………………………………………………………..…………………………..13

2.6. Tables…………………………………………………………………………………….………………………..15

3. Results and Discussion……………………………………………………………………………………..…….16

3.1. Granular Flow in a Shallow Bed…………………………………………………………………….…16

3.1.1. Torque and Power Measurements………………………………………….…………….16

3.1.2. Fluctuation Analysis of Torque Data…………………………………….……………….18

3.1.3. Fast Fourier Transform of Torque Data………………………….……………………..19

3.2. Effect of Impeller Blade Configurations……………………….............……………………….20

3.2.1. Effect of the number of impeller blades……………………..…………………………21

ii

vi

3.2.2. Effect of the impeller blade angle……………………………………….………………..23

3.2.3. Effect of the impeller blade length………………………………………………….…….25

3.3. Effect of particle friction coefficient…………………………………………………………………26

3.4. Effect of impeller blade position in a deep granular bed………………………….………27

3.5. Effect of particle size and scale-up………………………………………………………………..…30

3.6. Effect of particle fill height and scale-up…………………………………………………….……32

3.7. Figures………………………………………………………………………………………………….…………38

4. Conclusions……………………………………………………………………………………………………………50

5. Reference………………………………………………………………………………………………………………54

vii

LIST OF FIGURES

Fig. 1. A schematic of the laboratory experimental set-up for the impeller torque and power

measurements.

Fig. 2. A schematic of the 4-bladed mixer used in the experiments.

Fig. 3. Impeller blade configurations used in the experiments: (a) standard impeller, 4

blades, 45 mm length, 135° angle; (b) 2 blades, 45 mm length, 135° angle; (c) 1 blade, 45

mm length, 135° angle; (d) 4 blades, 45 mm length, 90° angle; (e) 4 blades, 45 mm length,

150° angle; and (f) 4 blades, 37.5-mm length, 135° angle. The black squares in this figure

show an area of 1 × 1 cm2.

Fig. 4. Normalized torque (�⃗� ∗) fluctuation profiles at steady state for different blade

rotational speeds: (a) 10 RPM, (b) 20 RPM, (c) 60 RPM, and (d) 200 RPM. The normalized

torque values shown in this figure are the instantaneous torques divided by the time-

averaged torque at each shear rate. Torque signal measurements were recorded every 0.125

s.

Fig. 5. Single-sided amplitude spectrums of the fast Fourier transform (FFT) torque signals

for different blade rotational speeds: (a) 10 RPM, (b) 20 RPM, (c) 60 RPM, and (d) 200

RPM. Torque signal measurements were recorded every 0.125 s.

Fig. 6. Granular flow properties as a function of the dimensionless shear rate (γ*) for “the

standard case”: (a) the time-averaged impeller torque and (b) the average adjusted power

(P*). The experiments were performed using red glass beads of 2-mm diameter loaded to a

30-mm fill height (shallow bed, H/D = 0.30) and the standard impeller blades (4 blades

with a 135° angle and 45 mm length). The adjusted power (P*) was obtained by subtracting

viii

the power draw at a specific RPM for the empty mixer from the power draw for the impeller

in the mixer loaded with particles.

Fig. 7. Effect of the number of impeller blades on: (a) the average torque and (b) the

average adjusted power as a function of the dimensionless shear rate. Impeller blades used

in these sets of experiments have 135° angle and 45 mm length.

Fig. 8. Effect of the impeller blade angle on: (a) the average torque and (b) the average

adjusted power as a function of the dimensionless shear rate. Impeller blades used in these

sets of experiments have 4 blades and 45 mm length.

Fig. 9. Effect of the impeller blade length on: (a) the average torque and (b) the average

adjusted power as a function of the dimensionless shear rate. Impeller blades used in these

sets of experiments for both cases have 4 blades and 135° angle.

Fig. 10. Effect of the macroscopic friction coefficient (µ) of particles on: (a) the average

torque and (b) the average adjusted power as a function of the dimensionless shear rate.

Experiments were performed using the standard impeller blades and the glass beads with a

diameter of 2 mm loaded to a 30-mm fill height (H/D = 0.30).

Fig. 11. Effect of the impeller blade position in a deep granular bed (90-mm fill height) on:

(a) the time-averaged torque and (b) the average adjusted power as a function of the

dimensionless shear rate, compared to those values in a shallow bed (30-mm fill height just

covering the top tip of the blades). Blade position in a deep bed: top (H/D = 0.30), middle

(H/D = 0.60), and bottom (H/D = 0.90). Experiments were performed using the standard

impeller blades and the glass beads with a diameter of 2 mm.

Fig. 12. Effect of the particle size on: (a) the average torque and (b) the average adjusted

power drawn from experiments as a function of the shear rate γ°. Experiments were

ix

performed using the standard impeller blades, and the granular bed was filled up to a 30-

mm height (H/D = 0.30) for all particle sizes.

Fig. 13. (a) The time-averaged normalized torque, ⟨𝑇⟩/𝑑, and (b) the average normalized

adjusted power, P*/d, plotted as a function of the shear rate γ°. Experiments were performed

using the standard impeller blades, and the granular bed was filled up to a 30-mm height

(H/D = 0.30) for all particle sizes. Note that the y-axis in Fig. 13b is multiplied by a factor

of 103.

Fig. 14. Effect of the amount of materials in a bladed mixer, reported as the H/D ratio, on:

(a) the time-averaged torque and (b) the average adjusted power as a function of the

dimensionless shear rate. Experiments were performed using the standard impeller blades

and the glass beads with a diameter of 5 mm. The effective H/D ratios indicate usage of a

lead weight instead of using actual particle mass.

Fig. 15. (a) The time-averaged torque ⟨�⃗� ⟩ and (b) the average adjusted power P* values in

Fig. 14 were normalized by square of the height (H2) of the granular material filled in a

bladed mixer plotted as a function of the dimensionless shear rate. Experiments were

performed using the standard impeller blades and the glass beads with a diameter of 5 mm.

The effective H/D ratios indicate usage of a lead weight instead of using actual particle

mass. Note that the y-axis in Fig. 15b is multiplied by a factor of 103.

1

1 INTRODUCTION

Mixing is a unit operation that is widely used in a variety of production processes ranging

from catalytic to chemical, cosmetic, food, and pharmaceutical industries [1,2]. In

pharmaceutical manufacturing, mixing is one of the steps of critical importance that

needs to be carefully controlled. Homogeneity of a mixture of excipients and an active

pharmaceutical ingredient (API) needs to be achieved to ensure that the drug

concentration in a finished product is consistent and that all dosage units produced are

of uniform potency [3]. There are several different types of mixers (also called blenders)

used in the pharmaceutical industry. Examples of mixers include rotating drums [1],

bladed mixers [3], high-shear mixers [4], agitated filter beds/dryers [5,6], and fluidized

beds [7,8]. Bladed mixers and high-shear mixers have advantages that they are simple in

construction and that they take a relatively short time to complete the mixing operation

[9]. The degree of uniformity of dosage units and the resulting quality of finished products

are functions of the material properties, the design of the mixer, and the operating

conditions [10–12]. Process monitoring and control are therefore important to maintain

stable product quality and improve process efficiency.

Monitoring of the mixing process can be carried out by various methods, such as

employing the particle image velocimetry (PIV) technique [13], measuring agitation

torque, and recording power consumption of the mixer [9]. While getting mixed, solid

particulate matter exerts a load on agitator blades and vice versa, thus, giving rise to the

2

agitation torque [14]. Changes in powder characteristics and flow behaviors in the mixer

are reflected by changes in the value of the impeller torque, e.g., when the endpoint of

granulation is reached [9]. The torque required to turn an impeller in a particle bed

depends on a number of factors, including blade design, agitator rotational speed,

material fill level, and mixer size [15–17]. In the case where agglomeration is being carried

out in a high shear mixer, torque measurement helps in control and monitoring of the

granulation process [18–20]. Since torque is sensitive to changes in operating conditions,

with rapid measurement and good precision, it can be used as a parameter to probe the

efficiency of mixer designs [18]. In addition, impeller torque measured from a laboratory

scale mixer can be used to characterize and predict the risk for agglomeration in powder

beds with varying degrees of moisture content at industrially relevant scales. Bulk friction

coefficient can also be measured for wet powder beds using torque data [21].

Impeller torque measurements can be implemented to monitor not only mixing and

agglomeration/granulation processes, but also agitated drying processes. Drying of API

crystals can be carried out in an agitated filter-bed dryer, which has advantages over tray

dryers as it reduces operation time by improving heat and mass transfer. In an agitated

dryer, torque data can provide a correlation between shear stress experienced by a

powder bed and the degree of attrition of particles [5]. Impeller torque at the laboratory

scale can also be used to obtain an estimate of the amount of work done per unit mass

by the impeller at larger scales. This relationship can help improve scale up of the agitated

drying process [5]. The ability to predict attrition and to obtain uniform size distribution

3

of API particles results in better critical quality attributes of the finished dosage forms

including content uniformity, dissolution rate, bioavailability, and stability.

Besides the impeller torque measurement, recording power consumed by the impeller

motor can also be employed to monitor and control mixing processes. Previous research

by Ritala et al. [22] demonstrated that power consumption during rotation of the blades

was a function of intra-granular porosity and surface tension of a binder solution in wet

mixing operations. Additionally, a correlation existed between power consumption and

the strength of the moist agglomerates. An increase in cohesiveness or tensile strength

of the moist granule mass resulted in a change in power consumption, indicating that the

liquid solution characteristics, such as surface tension and contact angle, had an influence

on power consumption [22]. Jirout and Rieger [23] illustrated that, in the case of

suspending solids in liquid systems, the power consumption required for off-bottom

suspension of the solid particles could be used to compare the efficiency of different

types/configurations of impellers for the mixing operation.

Although power consumption measurement has been extensively carried out in previous

research on the liquid mixing process, there has been limited work for monitoring mixing

of solids systems. With the current interest of moving from batch to continuous

manufacturing in the pharmaceutical industry, continuous mixing processes have been

extensively studied in recent years [24–26].

4

Vanarase et al. [27,28] conducted experiments on a continuous powder mixer to examine

the effects of operating conditions, design parameters, and material flow properties of

pharmaceutical mixtures as a function of shear rate. Relative standard deviation (RSD),

one of the most common mixing indices, was used to characterize mixer’s performance

and in turn the efficiency of the mixing process. RSD is a measure of blend homogeneity,

and for a continuous process they found RSD depended on several factors such as blade

rotation rate, material flow rate, and mixer blade configuration. They demonstrated that

the number of blade passes a measure of the strain, was a function of the blade rotational

rate. Although there is interest in the pharmaceutical industry in continuous

manufacturing, most products are still manufactured in batch mode. Thus, there is still a

need to better understand batch operations.

Campbell [29,30] and Campbell and Brennen [31] outlined a theoretical method for

calculating collisional stresses within a granular bed from the total contact force between

particles, particle diameter, and volume of a sampling cell. The important stress tensor

components in the area of granular material flows are the average normal stress and the

shear stress. Knowing the normal stress values, the pressure inside a granular bed can be

calculated. According to a granular rheological model [32,33] a number of experimental

and numerical studies suggested that the mass density and the macroscopic (bulk) friction

are functions of the inertial number, which can be computed from the shear rate, the

particle diameter and density, and the compressive pressure. Stresses and torque can be

measured from the macroscopic friction coefficient and the inertial number. In previous

5

work, Cavinato et al. [34] conducted experiments on the mixing and flow behaviors of

cohesive pharmaceutical powders in lab-scale and pilot-scale high shear mixers. They

proposed a scaling relationship based on a dimensionless torque number which was a

function of the mass fill and the square root of the impeller Froude number. It was found

that the mass fill was one of the important parameters that had an impact on the powder

flow patterns during the high shear mixing process.

In spite of some previous research in this area, the roles of material properties, blade and

equipment configurations, and operating parameters on the impeller torque and power

consumption in a batch mixer are still poorly understood. In this thesis, we studied a bed

of particles in a batch bladed mixer consisting of a cylindrical glass vessel and a vertical

impeller blade. Particle properties and impeller blade configurations of the mixing system

were varied to investigate their effects on the experimentally measured torque that the

impeller exerted on particles and on the power consumption that the motor drew to

move the blades through the granular bed.

6

2 MODEL AND EXPERIMENTAL SET UP

2.1 Experimental set-up and mixer geometry

The laboratory equipment used in this work is shown in Fig. 1. The unit was composed of

a bladed mixer, a signal transducer with a load cell, a data recorder with a monitor, and a

power meter. The unit is similar to what was used in previous work [5]. A cylindrical glass

vessel and Teflon impeller blades were purchased from Chemglass Life Sciences©. The

glass chamber was affixed to a rotating table with a metal lever arm which applied force

on a load cell when the blades moved through the particle bed. In the experiments, the

impeller blades were attached to one end of a cylindrical stainless steel agitator shaft

which was connected to a motor at the other end. The motor was firmly held on a

Chemglass© bench top support stand and wired to a digital motor controller which was

used to adjust the direction and the speed of rotation of the impeller blades. The data

acquisition device used in this research was a Yokogawa™ DX1006-1. It was employed to

record voltage signals to monitor the blade rotation speed and the force applied on the

load cell (from which torque can be calculated). The power meter used in this study was

a Vernier Software & Technology© Watts Up PRO; it was connected to the motor

controller to measure power consumed by the impeller blades when rotated through a

granular bed.

A schematic of the mixer geometry used in this work is depicted in Fig. 2, and the

dimensions of the bladed mixer are reported in Table 1. The cylindrical vessel has an inner

7

diameter (D0) of 100 mm and a total internal height of 155 mm, which can hold a volume

of approximately 1.2 liters. The mixer is mechanically agitated by impeller blades, and the

agitator shaft is placed at the center of the cylindrical mixer. The diameter of the

cylindrical impeller shaft (D1) is 10 mm, and H0 is the height above the blades which is 128

mm for our standard setup. The vertical span of the impeller blades (H1) is 25 mm, and

the length (L) of each blade measured from the center of the impeller shaft to the edge

of the blade is mm for the initial impeller blades used in this work. The clearance

between the bottom of the blades and the base of the mixer (H2) is set to approximately

2 mm in all experiments.

2.2 Materials

The granular material used in this research was Dragonite® composite glass beads (Jaygo

Incorporated, Randolph, NJ) with a density of 2500–2550 kg/m3. Physical properties of

the cohesion-less spherical glass beads used in the experiments are listed in Table 2. Three

different diameter sizes of the beads were utilized in the experiments: 1 mm (0.60–1.40

mm), 2 mm (1.70–2.36 mm), and 5 mm (4.71–5.01 mm). Particle size distribution analysis

of the glass beads was conducted using sieve analysis, and the ranges for the three

different particle diameters were reported in the parentheses above. The glass beads of

the 1-mm diameter were colorless, and the 5-mm particles were surface-colored purple.

Two types of 2-mm glass beads were used in the experiments: surface-colored red beads

and clear beads. The 2-mm red glass beads were used for our initial experiments while

8

the 2-mm colorless glass beads were used for subsequent particle surface roughness

modification experiments which will be described later in Section 2.4. It was noticed that

after running the experiment for some time, particles started to stick to the walls of the

vessel. An anti-static ionization blower and anti-static spray were therefore used to

reduce static charges [35] and to produce reliable and repeatable results.

2.3 Impeller torque and power measurement protocol

Before performing each experiment, the impeller shaft was first placed at the center of

the bladed mixer, and then glass beads were gently loaded into the mixer and allowed to

settle while the blades remained stationary. Blade movement was then started, and the

signal measurement and recording were commenced. The impeller blades were rotated

in the counter-clockwise direction leading to an obtuse-angle blade orientation which is

a commonly used configuration in industry. Experiments were carried out at various blade

rotational speeds ranging from 10 to 200 revolutions per minute (RPM). At these shear

rates, granular flows in the bladed mixer occurred in the quasi-static and intermediate

regimes according to the flow regime map proposed by Tardos et al. [36]. When starting

the blade movement, the rotational speed of the impeller was gradually increased from

0 RPM to a desired rotation rate for each experiment. In this research, the signal data was

measured and recorded once every second for the time-averaged torque and every 0.125

seconds for the torque fluctuation analysis. The total time duration for recording signals

for each individual experiment was 3 minutes and 30 seconds, and for each blade

9

rotational speed the experiment was repeated three times. Mean and standard deviation

of instantaneous data at each time step were calculated from the three separate data

sets. In general, the steady state of a system was usually reached within 30 – seconds

of the blade movement for all experiments. Once the steady state was reached, the time

averaging procedure for the mean torque and power readings was achieved throughout

the measurement period.

The torque was measured by measuring a force and multiplying this by the lever arm

length. In this set-up, force was measured by using a load cell attached to a metal arm

that pivots around a rotating table in which the glass vessel assembly sits, as shown in Fig.

1. As the impeller moves through the granular bed, the blades transmit force to the

particle bed causing the material to rotate relative to the mixer wall. The rotation of the

glass vessel causes the pivot arm to apply force on the load cell which measures the force.

With this set-up, torque can be calculated from cross-multiplying the length of the lever

arm and the measured force:

�⃗� = ℓ × 𝐹 (1)

where �⃗� is the torque acting on the load cell, ℓ is the lever arm length, and 𝐹 is the

measured force that the lever arm acts on the load cell.

10

It can be shown that the torque exerted on the impeller at the shaft equals the torque

exerted on the mixer (walls) due to the moving granular bed [37]. Knowing the agitation

torque, the average shear stress experienced by the granular bed can be computed from

the equation given by Darelius et al. [5,38]:

⟨𝜏𝜃𝑟⟩ =�⃗�

2𝜋𝑅cyl2 𝐻m

(2)

where ⟨𝜏𝜃𝑟⟩ is the average shear stress, �⃗� is the measured impeller torque, Rcyl is the

radius of the cylindrical glass mixer, Hm is the height of the moving particle bed when the

blades are rotating. Moreover, work done by the impeller can be calculated from the

impeller torque by using the following expression:

𝑊 = ∫ �⃗� 𝑡

0∙ 𝜔𝑑𝑡 (3)

where W is the work done by the impeller blades over time t, �⃗� is the impeller torque,

and ω is the angular velocity of the impeller blades.

As described above, the torque measured in our experiments is the torque needed to

move the granular bed. However, for power consumption measurement, the power is

made up of two components. One component is the power needed to run the motor, and

the other is the power needed to move the particle bed. The power needed to run the

11

motor is therefore subtracted from the actual power reading measured by the power

meter when a mixer is loaded with particles:

P* = P - Pemp (4)

where P* is an adjusted power for the charged cylinder for a given RPM, P is the actual

power measured by the power meter for the charged cylinder for a given RPM, and Pemp

is the power measured for an empty mixer at that same RPM. All powers are measured

in watt (W).

2.4 Particle surface roughness

A method for modifying the surface of glass beads is described by Remy et al. [39], and

this protocol was adopted in this study in order to investigate the effect of particle surface

friction on the impeller torque and power. In this procedure, a rotating drum was

employed to coat the surface of 2-mm colorless glass beads with magnesium stearate

(MgSt). Magnesium stearate was chosen as a coating material due to the increased

asperity and the irregular shape of its primary particles. The adhesive interaction between

MgSt and the surface of the glass beads occurred to physically coat the glass beads with

MgSt. The magnesium stearate was purchased from Sigma-Aldrich® (CAS No. 557-04-0)

and was composed of irregular plates with a mean diameter of 15 µm. The rotating drum

used as a coater was made out of glass with an outer diameter of 20.5 cm and a length of

12

8 cm. The rotating drum contained 4 Teflon baffles equally spaced along the side wall of

the drum. The baffles were 3 cm deep and 7 cm in height. The protocol for particle surface

roughening is briefly mentioned as follows. In total 200 – 400 g of the beads were added

into the rotating drum, followed by 50 – 100 g of MgSt to achieve a 25% w/w MgSt

concentration. The drum was rotated at 30 RPM under atmospheric pressure for 15

minutes. Some small MgSt agglomerates were created during the coating procedure. The

resulting mixture was sifted through No. 8 and No. 12 sieves in order to separate the

coated beads from the MgSt agglomerates. The resulting MgSt-coated glass beads had a

particle diameter ranging from 1.7 mm to 2.4 mm, a size range similar to that of the

uncoated glass beads (see Section 2.2). Using a Freeman Technologies FT4 rheometer, a

macroscopic/bulk friction coefficient (µ) was measured. For the 2-mm MgSt-coated glass

beads the macroscopic friction is 0.38 ± 0.008 compared to 0.32 ± 0.009 for the uncoated

beads [39]. MgSt is universally known to be a lubricant for pharmaceutical powders when

it is added to pharmaceutical powders in small quantities (up to around 1.5% w/w). At the

same time, it has been observed that when added in large quantities (around 25% w/w)

to glass beads, it serves to increase the macroscopic friction coefficient of the glass beads

[39]. This is because at a particle surface level MgSt-coated beads have some surface

roughness and irregularity due to the increased asperity and the irregular shape of the

MgSt primary particles while the uncoated glass beads have a fairly smooth surface.

Results for the effect of particle surface roughness on measured impeller torque and

power consumption will be discussed in Section 3.3.

13

2.5 Figures for Chapter 2

Fig. 1. A schematic of the laboratory experimental set-up for the impeller torque and power

measurements.

Fig. 2. A schematic of the 4-bladed mixer used in the experiments.

D0

D1

H0

L

H1

H2

14

Fig. 3. Impeller blade configurations used in the experiments: (a) standard impeller, 4

blades, -mm length, 135° angle; (b) 2 blades, 45 mm length, 135° angle; (c) 1 blade, 45

mm length, 135° angle; (d) 4 blades, 45 mm length, 90° angle; (e) 4 blades, 45 mm

length, 150° angle; and (f) 4 blades, 37.5-mm length, 135° angle. The black squares in this

figure show an area of 1 × 1 cm2.

15

2.6 Tables for Chapter 2

Variable Symbol

D0 100

D1 10

L

H0 128

H1 25

H2 2

Table 1: Laboratory mixer dimensions.

Variable Symbol Value

Particle diameter d 1, 2, and 5 mm

Particle density ρp 2.50–2.55 g/ml

Macroscopic friction

coefficient (uncoated beads)

µ 0.32

Mass of granular bed m 300–900 g

Table 2: Glass bead properties used in experiments.

16

3 RESULTS AND DISCUSSION

3.1 Granular Flow in a Shallow Bed

In this section, we first consider granular flows in a shallow bed, which we will refer to as

“the base case”. In our base case the 2-mm monodisperse, cohesion-less, spherical,

uncoated red glass beads were used. The amount of a granular material loaded in the

mixer was approximately 300 g, which was sufficient to cover just the top tip of the

impeller blades, and the initial fill height of the stationary particle bed, H, was measured

as 30 mm (H/D = 0.30). Experiments were performed in the bladed mixer with dimensions

given in Table 2. The impeller (Fig. 2) used in experiments in this part consisted of 4 blades

pitched at 135° angle with respect to a horizontal plane, and each blade had a length of

mm, which we will consider as “the standard impeller blades”.

3.1.1 Torque and Power Measurements

The protocol described in Section 2.3 was followed, and the torque signals were

measured and recorded every second for experiments in this section. The time-averaged

torque and the average adjusted power are plotted as a function of the dimensionless

shear rate (γ*) as shown in Fig. 4.1. The dimensionless shear rate (γ*) is defined by the

following equation according to Tardos et al. [36]:

γ ∗ = γ°√d

g (5)

17

where γ° is the shear rate (s–1), d is the diameter of particles (m), and g is the gravitational

acceleration (m/s2). The shear rate (γ°) can be computed from the blade rotational speed

(in RPM) by using the formula:

γ° = 2π ×Rotational Speed

60 (6)

In Fig. 4a, it can be seen that the average torque increases linearly with the blade

rotational speed. One may expect that there would be two granular flow regimes as

described by Tardos et al. [36]. The first one is the quasi-static or slow regime (γ* is smaller

than ~ 0.1). The second regime is called the intermediate regime (γ* >~ 0.1). In the quasi-

static regime one would expect the torque to be independent of rotation rate. We do not

observe such a regime for a shallow particle bed (H/D = 0.30). We will come back to this

point when we examine deep beds as in that case we do observe a quasi-static regime

where torque is independent of shear rate. The trend of the linear increase in torque as

a function of shear rate in the intermediate regime observed from experimental results

in this section are qualitatively similar to numerical results from previous research studies

using a discrete element method (DEM) computational technique done by Chandratilleke

et al. [15] and Sato et al. [9].

Fig. 4b displays a graph of the average adjusted power plotted as a function of the

dimensionless shear rate. The adjusted power (P*) was obtained by subtracting the power

18

consumed at a specific RPM with an empty cylinder from the power consumed by the

impeller in the mixer loaded with particles. Similar to the torque graph in Fig. 4a, the

average adjusted power linearly increases for the entire range of the dimensionless shear

rate. In this research, it is assumed that the power needed to run the motor under load

can easily be separated from the power needed to move the impeller by subtracting the

power needed to run the motor when the mixer is not under load (i.e., when the mixer is

empty and has no particles charged). However, this assumption might not be always true

for all settings. Nevertheless, the power consumption measurement still provides useful

information for pharmaceutical and other industries dealing with mixing processes, but

care needs to be taken since recording power data is not as easily interpreted as

measuring torque.

3.1.2 Fluctuation Analysis of Torque Data

The torque signals were measured and recorded every 0.125 seconds for all the

experiments in this part. The instantaneous torque values ⟨T⃗⃗ ⟩ measured at each time step

were normalized by the time-averaged torque, ⟨T⃗⃗ ⟩ at each blade rotational speed:

T⃗⃗ ∗ =T⃗⃗

⟨T⃗⃗ ⟩ (7)

where T⃗⃗ ∗ is the dimensionless normalized torque at each time step. Fig. 5a – d display the

normalized torque T⃗⃗ ∗ fluctuation profiles after the particulate systems have reached a

19

steady state for different blade rotational speeds: 10 RPM, 20 RPM, 60 RPM, and 200

RPM, respectively. The time span shown in these graphs is 25 seconds and was sampled

from the middle period of the experiments at steady state. The smallest fluctuation

amplitudes can be seen in the case where the impeller blades rotate at 10 RPM (Fig. 5a).

The torque fluctuation profiles for 10 RPM (Fig. 5a) are comparable in amplitude to those

for 20 RPM and 60 RPM (Fig. 5b – c, respectively). As the shear rate increases in the

intermediate regime of granular flow, i.e., increasing the blade rotational rate from 60

RPM (Fig. 5c) to 200 RPM (Fig. 5d), the fluctuation amplitudes increase. At 200 RPM (Fig.

5d), there is a noticeable recurrence of a periodic burst behavior of the torque

fluctuations, indicating low and high values of torque which we conjecture are due to the

expansion and compression states of the granular bed in the highest shear rate case of

our experiments.

3.1.3 Fast Fourier Transform (FFT) of Torque Data

We also computed the discrete Fourier transform of the torque signals using the fast

Fourier transform (FFT) algorithm in MATLAB® for the same blade rotation rates

mentioned in Section 3.1.2. Torque data input for the FFT was measured and recorded

every 0.125 seconds. Single-sided amplitude spectrums of the FFT torque data are

depicted in Fig. 6 for various rotational speeds of the impeller blades corresponding to

Fig. 5. As a check of the algorithm, we confirmed that the values of the 0-Hz components

(the peaks at 0 Hz) always corresponded to the time-averaged torques for each blade

20

rotation speed. Peaks corresponding to the impeller blade rotational rates (in RPM) can

be observed (as circled for each rotational speed in Fig. 6), and the frequency (f) of each

peak in Hz can be calculated by the following formula where the rotational speed is in

RPM:

𝑓 =𝑅𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑆𝑝𝑒𝑒𝑑

60 (8)

For example, the peak at frequency of 0.167 Hz can be seen in Fig. 6a for the impeller

blades rotating with 10 RPM. The height of the blade frequency peaks increases with an

increase in the rotational speed of the impeller blades. It is interesting to note that there

are several overlapping peaks occurring between 2 – 3 Hz for all shear rates. We

hypothesize that these peaks between 2 – 3 Hz are associated with natural frequencies

of the moving granular material itself. It is also important to state that in all cases these

peaks are higher in amplitude than the rotating blade frequency peaks.

3.2 Effect of Impeller Blade Configurations

A photo of the different types of the impeller blades used in our experiments is presented

in Fig. 3. The black squares in this figure show an area of 1 × 1 cm2. Fig. 3a is what we

referred to as “the standard impeller blades”. In this part we explore the effects of the

impeller configurations on the measured torque and power by changing the number, the

angle, and the length of the blades using the impellers in Figs. 3b – f.

21

3.2.1 Effect of the Number of Impeller Blades

DEM simulations have been used by Boonkanokwong et al. [40] to study the effect of the

number of impeller blades on granular flow behaviors and mixing kinetics in an agitated

mixer. Different numbers of the blades in a mixer resulted in different impeller contact

forces and particle velocity profiles, and thus, different mixing performance. The number

of the blades used in a mixing system can also affect the impeller torque exerted on

granular materials in a mixer and the equipment power consumption. This will be

experimentally investigated further in this section. Experiments were carried out with

different sets of impeller blades used in a mixer: 4 blades, 2 blades, and 1 blade as shown

in Figs. 3a – c, respectively. All of these impeller blades are pitched at a 135° angle and

have the same length (L) of 45 mm. The 2 and 4 blades are symmetrically placed about

the vertical axis of the impeller shaft. All other particle parameters and operating

conditions were the same as in the base case discussed in Section 3.1.

In Fig. 7a, the time-averaged torque for different blade numbers is plotted as a function

of the dimensionless shear rate. The trends of how torque behaves as a function of the

impeller blade rotational rate for the 1- and 2-bladed mixers are similar to that for the 4-

bladed mixer (see Section 3.1.1). Interestingly, the average torque measured in the 2-

bladed mixer is larger than that in the 4-bladed mixing system at all blade rotational

speeds, and the value measured from the single-bladed mixer is significantly less than

that of the 2 and 4 blades cases. In previous work [40], it was found that higher radial and

vertical velocities of particles were observed in the 1- and 2-bladed mixers, which led to

22

more pronounced three-dimensional recirculation patterns. However, the tangential

velocity components of particles in the 3- and 4- bladed cases were larger. Additionally, it

was found that using two or three impeller blades provided better mixing performance

than using one or four blades, as evaluated by calculation of the relative standard

deviation (RSD) and the Lacey index of the systems. Granular temperature and particle

diffusivities obtained for the 2- and 3-bladed cases were also higher than those for the 1-

and 4-bladed mixers. Solids fraction analysis showed that dilation of the particle bed

occurred to the greatest extent in the 2-bladed mixer. In our work we observe that the

average torque measured in the 2-bladed mixer is larger than that in the 4-bladed mixer

at all blade rotational speeds, and the value measured from the 1-bladed mixer is

significantly less than those of the 2 and 4 blades cases. We hypothesize this is due to the

fact that in the 1-bladed mixer the averaged tangential contact force that the impeller

blades exert on particles is less than that in the 4-bladed mixer which is in turn less than

that in the 2-bladed mixer [40]. The average adjusted power is presented in Fig. 7b, and

it can be seen that the 1-bladed mixer has lower values than that of the 2- and 4-bladed

mixers. In addition, the average adjusted power for the 2-bladed case is higher than the

4-bladed case which is in agreement with the torque measurements (see Fig. 7a). The

trend of a linear increase of the adjusted power with an increase in the shear rate is

observed for all numbers of blades used in this section. The fluctuations of both torque

and power values at each rotation rate in the 1-bladed mixer are generally higher than

those in the 2- and 4-bladed cases because of asymmetry of the single blade when

rotating through a granular bed.

23

We should note that both the torque and power that we measure go to zero when the

rotational speed is zero. At the same time, for both Fig. 7a and b, if you extrapolate the

data back to a zero rotational speed, it looks like there will be a finite value of torque and

power at zero rotational speed. This is because we are not able to collect data at very low

dimensionless shear rates; we cannot go below a dimensionless shear rate of around 0.01

in Fig. 7. This is due to the fact that we cannot get our motor to move the impeller through

the granular bed at very low rotation rates. Our motor does not have enough

power/torque at low rotation rates to start moving, and it therefore becomes stuck in the

granular bed. Further work would be needed with a more sophisticated motor that can

operate at very low rotation rates, to discern the exact behavior at low rotation rates.

Moreover, we expect complex behavior at very low rotation rates including stick-slip

phenomena.

3.2.2 Effect of the Impeller Blade Angle

We next explore the effect of varying the impeller blade angle on granular flow

properties. For the standard impeller (Fig. 3a), blades are pitched at a 135° angle with

respect to the horizontal plane. The angles of the blades used for the experiments in this

section were altered to 90° (Fig. 3d) and 150° (Fig. 3e). The impellers were set to rotate

in the counter-clockwise direction to provide obtuse blade angle configurations, which

are commonly used in industries and considered to result in better mixing performance

than the acute angles [41]. All other particle parameters and operating conditions were

24

the same as in the base case. Time-averaged torque and average adjusted power plotted

as a function of the dimensionless shear rate are displayed in Fig. 8. From Fig. 8a, it can

be seen that the blades with 90° angle have the highest average torque values, followed

by 135° and 150° angles, respectively. In general, the torque values for the 135° and 150°

cases are quite similar.

In Fig. 8b, it is seen that the average adjusted power readings for the three different blade

angle configurations are closer to one another than the torque values (see Fig. 8a). We

observe in general less differences in power than in torque. We hypothesize that this is

because measuring power consumption is more complicated than measuring the torque

since power is composed of two components (the power needed to run the motor and

the power needed to move the granular bed). We wanted to report values for the power

required to move the particle bed (i.e., the adjusted power). We therefore measured the

total power drawn by the motor and computed the adjusted power by subtracting out

the power needed to run the motor when the mixer is empty from the total power meter

readings (at the same impeller blade rotational speed). We make the assumption that the

power needed to run the motor when the mixer is charged with particles is equal to that

when the mixer is empty, which might not be true for all settings and may partly

contribute to the fact that there is no significant effect of different blade angles on the

adjusted power at low impeller rotational speeds. In general, the average torque and the

adjusted power decrease with an increase in the impeller blade angle. Our experimental

data is in good agreement with the DEM simulation results accomplished by

25

Chandratilleke et al. [15], examining the effect of rake angle of the impeller on the shaft

torque for cohesive particles in a vertical bladed mixer.

3.2.3 Effect of the Impeller Blade Length

We next investigate the effect of the length (L) of blades by using two different types of

impeller configurations in our experiments: the -mm standard blades (Fig. 3a) and the

37.5-mm blades (Fig. 3f). Both impellers have four blades with a 135° angle. All other

particle parameters and operating conditions were the same as in the base case. In Fig.

9a, it is noticed that when the time-averaged torque is plotted as a function of the

dimensionless shear rate, torque values for both blade lengths increase fairly linearly with

shear rate. In addition, the time-averaged torque of the 45 mm blades is significantly

higher than that of the 37.5-mm blades for the entire range of impeller rotational rates.

It can be observed in Fig. 9b that the average adjusted powers at low shear rates are not

obviously different for the two blade lengths. We hypothesize that this is for similar

reasons to what was discussed earlier for Fig. 8b, i.e. measuring the power consumption

is complex in nature. We have assumed that the power needed to run the motor when

the mixer is not under load (empty mixer) equals the power needed to run the motor

when the motor is under load (charged mixer). As we discussed previously, this

assumption may not be true for all conditions and may contribute to the adjusted power

results being similar especially for low rotational speeds of the impeller blades. However,

26

at higher shear rates as the dimensionless shear rate increases, the average adjusted

power is higher for the 45 mm standard blades than the 37.5-mm blades.

3.3 Effect of Particle Friction Coefficient

Remy et al. [39] experimentally and computationally studied the effect of varying

roughness of particle surface on granular flow behaviors and mixing kinetics. They found

that particle friction had a tremendous impact on particle velocity profiles and mixing

patterns of granular materials in a bladed mixer. In this section, the effect of particle

surface roughness on impeller torque and power will be investigated. The experiments

were conducted for two sets of 2-mm glass beads; one was the standard uncoated red

beads, and the other was the magnesium stearate (MgSt)-coated colorless beads

fabricated according to a protocol in Section 2.4.

The fill height of the particle bed for both cases was 30 mm (H/D = 0.30). The standard

impeller blades (Fig. 3a) were used, and all other operating conditions were as in the base

case. The results for the effect of roughness of the particles on the measured torque and

adjusted power are shown in Fig. 10. It is observed in Fig. 10a that the MgSt-coated beads

(m = 0.38, see Section 2.4) exhibit a distinctively higher average torque than the uncoated

beads (m = 0.32, see Section 2.4). The reason for this is that the coated beads have a

higher friction than the uncoated beads, and thus the impeller blades require more force

to move through the particle bed. Also, the standard deviations of the torque values at

27

each rotational rate are more for the MgSt-coated beads than those in the regular bead

case as shown by the magnitude of the error bars in Fig. 10a. As shown in Fig. 10b, the

average adjusted power for the uncoated beads is noticeably less than that of the MgSt-

coated beads for the entire range of the impeller blade rotational speed. These trends of

torque and power consumption as a function of shear rate are similar to results for an

experimental study performed by Remy [42] and DEM simulations carried out by Benque

[43].

3.4 Effect of Impeller Blade Position in a Deep Granular Bed

Remy et al. [41] demonstrated via DEM computational studies that granular flows in a

shallow bed behaved differently from those in a deep particle bed. In this part, we will

experimentally compare the measured agitation torque and the power consumption for

shallow and deep beds. The granular material used in this section’s experiments was 2-

mm red colored glass beads, and the amount of material loaded into the laboratory mixer

was increased from an initial bed height of 30 mm (as in the base case) to 90 mm.

It should be noted that we observed in our experiments that when the blades are rotating

the total height of the granular bed is approximately the same as the granular bed initial

height at a stationary state. The particle bed with a 30-mm fill height (H/D = 0.30), i.e.,

the amount of glass beads just covering the top tip of the blades, was considered to be

the shallow bed, and the particle bed with a 90-mm fill height was considered to be the

deep bed.

28

The H/D ratios in the deep bed are calculated from the height of the glass beads that

cover and are above the span of the impeller blades, i.e., H is defined as the height from

the bottom of the blades to the top of the granular bed. This was done as it had been

shown previously that the material above the blades affected torque measurements [41,

]. In order to investigate this impact, the blade position in a deep granular bed was

changed from the bottom (H/D = 0.90) to the middle (where the blades are at 30-mm

height from bottom, H/D = 0.60) and to the top (where the blades are at 60-mm height

from bottom, H/D = 0.30) in the mixer, compared to results in the shallow bed (30-mm

fill height, H/D = 0.30). The standard impeller blades (Fig. 3a) were used, and all other

process parameters in the experimental set-up were the same as those in the base case.

Comparison between the shallow and the deep granular beds for the measured torque

and power plotted as a function of a dimensionless shear rate is shown in Fig. 11.

From Fig. 11a, one can see that the time-averaged torque for the deep bed cases for H/

D = 0.60 and 0.90 are considerably larger than that for H/D = 0.30. In addition, the deep

and shallow bed cases for H/D = 0.30 agree fairly well. For the deep bed cases of H/D =

0.60 and 0.90, we now see that for low shear rates there is a quasi-static region (for γ*

less than approximately 0.05) where the torque does not vary with shear rate. For the

intermediate regime (γ* larger than approximately 0.05), the torque values linearly

increase as the blade rotational speed increases for all cases in the deep bed. Depicted in

Fig. 11b, the average adjusted power for the deep bed increases with a relatively large

slope as the impeller rotation rate increases; whereas, the power for the shallow bed

29

increases with a relatively smaller slope as the blade rotational speed increases. From Fig.

11a and b, it can be seen that the time-averaged torque as well as the average adjusted

power for the blades at the bottom case (H/D = 0.90) are significantly larger than those

for the cases in which the blades are at the middle position (H/D = 0.60) and at the top

position (H/D = 0.30) in the deep granular bed, respectively. This is because as the amount

of materials on top of the blades increases, the hydrostatic pressure applied on the

impeller increases, and thus, the resulting shear stress also increases correspondingly (see

Eq. (10)).

It can be observed that the average torque and the adjusted power results for the blades

at the top position (H/D = 0.30) in the deep bed case are not significantly different from

those for the shallow bed case (30-mm fill height in which particles are just covering the

top tip of the blades with the impeller at the bottom of a mixer, H/D = 0.30). As was

discussed, it has been shown in previous work [41, ] that the materials underneath the

impeller blades do not have any substantial influence on flow of grains in a mixing system

since the only portion of a granular bed that moves is material within the span of the

blades or close to the blades. The H/D = 0.30 case for the shallow and deep beds (see Fig.

11) are not identical; the small differences may be due to the fact that while the impeller

at the top of a deep bed is rotating, the blades push some portion of the glass beads

downward and cause some particles below to move which affects the agitation torque

and power values.

30

3.5 Effect of Particle Size and Scaling Relationship

Particle size is one of the important material properties that plays a crucial role in granular

flow and mixing. To understand the effect of particle size on power and torque exerted

by an impeller blade on a granular material, three different particle sizes were examined

in our experiments: 1-mm, 2-mm, and 5-mm diameter glass beads. All experiments in this

section were performed using the standard impeller blades (Fig. 3a), and the fill height of

the particle bed was kept constant to be 30 mm (H/D = 0.30) for all particle size cases. In

addition, all other parameters and processing conditions were set to those in the base

case.

In previous sections, the average torque and power values were plotted as a function of

the dimensionless shear rate (γ *). However, when plotting in this manner, the data points

for different bead sizes do not lie on the same x-axis scale, making it more difficult to

interpret the results. This is because the dimensionless shear rate (γ*) is a function of

particle size itself (see Eq. (5)).

Therefore, in order to fit data points on the same x-axis scale, for this section only the

time-averaged torque, as well as the average adjusted power, are plotted as a function of

the shear rate (γ°), as shown in Fig. 12. In Fig. 12a, the average torque measured for the

5-mm diameter particles is the highest, followed by the values for 2- mm and 1-mm beads,

respectively. A trend of linearly increasing torque values as the blade rotational rate

increases can be observed for the 1-mm particles, similar to the 2-mm case (the base case

31

in Fig. 4). At low shear rates (γ° < 4 s-1) fairly constant torques are observed for the 5-mm

beads indicating a quasi-static regime, and for higher shear rates there are linearly

increasing torque values (indicating an intermediate regime) as the shear rate (or the

blade rotational rate) increases. Similar trends are generally observed for the adjusted

power graph in Fig. 12b, but the values linearly increase with shear rate for all particle

sizes. Jop et al. and Andreotti et al. [46] have observed that granular flows can be

characterized by a dimensionless inertial number which is proportional to the particle

diameter, d, so it is possible that we may be able to scale the agitation torque by the

particle diameter and have the results collapse on top of another. In addition, the power

consumption of the motor can also be related to the torque (as in theory power is torque

times impeller blade rotational speed), and thus, we expect that we may also be able to

scale the power consumption by the diameter of particles.

In Fig. 13, we normalize the time-averaged torque and the average adjusted power values

in Fig. 12 by the particle diameter for the three different cases of 1-mm, 2-mm, and 5-mm

glass beads. It can be observed from Fig. 13a that the 1-mm particles have the highest

normalized torque⟨𝑇⟩/𝑑. It is interesting to note that, although the actual time-averaged

torque increases with an increase in a particle diameter (see Fig. 12a), the normalized

torque values⟨𝑇⟩/𝑑, for the 2-mm and 5-mm particles are lower than that of the 1-mm

particles. Moreover, the normalized torque ⟨𝑇⟩/𝑑 values for the 2-mm and 5-mm

particles are fairly similar, implying that the torque exerted by the impeller on the

granular material flowing in our bladed mixer complies with this theoretical scaling.

32

Discrepancy of the normalized torque values ⟨𝑇⟩/𝑑, for the 1-mm particles from the other

cases might be due to the relatively wide particle size distribution of the 1-mm glass beads

(see Section 2.2) and electrostatic charges that could extensively affect flow behaviors of

the 1-mm beads.

Similar trends are observed in Fig. 13b for the average adjusted power normalized by the

particle diameter, (P*/d), as a function of the shear rate. As mentioned above, the power

consumption of the motor can in theory be related to the torque since power is torque

times impeller blade rotational speed. We have examined if there is a simple relationship

between the torque and the adjusted power, and we have not observed a direct

relationship between the average torque values and the adjusted power. Once again, we

hypothesize that this is because we have assumed that the power needed to run the

motor when the mixer is not under load (empty mixer) equals the power needed to run

the motor when the motor is under load (charged mixer). As we discussed previously, this

assumption may not be true for all conditions and may contribute to the adjusted power

results not being simply related to the torque. Further work is needed to understand this

relationship.

3.6 Effect of Material Fill Height and Scale-up

Barczi et al. studied the effect of the granular bed depth on particle flow and

homogenization in a vertical bladed mixer with two flat blades using the DEM technique.

They observed that both macroscopic and microscopic flow patterns were affected by the

33

packed-bed depth and the blade rotational rate. Remy et al. carried out DEM simulations

of bladed mixers at different scales and found that the total weight of the particle bed in

a bladed mixer was an important parameter affecting stress profiles and granular flow

behaviors. In this part, we will examine the effect of the amount of granular material,

reported as the fill height in the mixer, on measured torque and power. To do so, 5-mm

glass beads and a lead weight were used in the experiments in this section to imitate the

hydrostatic pressure experienced by the impeller blades. A circular lead weight was

placed on a circular-cut piece of cardboard which was inserted through the impeller shaft

assembly. The lead weight and the cardboard sat on top of the glass beads and distributed

the normal load onto the particle bed during agitation to mimic the hydrostatic pressure

experienced in the mixer. The experiments were carried out for the following different

initial fill heights of beads: 30 mm (H/D = 0.30), 60 mm (H/D = 0.60), and 85 mm (H/D =

0.85). It was determined that mass of the lead weight (530 g) was equivalent to the mass

of a particle bed fill height of 44 mm. This information was used to increase the fill height

without actually using glass beads. For the 85-mm fill height, we performed two different

sets of experiments.

One scenario was using the beads actually filled up to 85 mm (H/D = 0.85), and the other

setting was using the glass beads filled up to 41 mm plus the lead weight (equivalent to

mm) on top (which we referred to as ‘‘Effective H/D = 0.85”). For the ‘‘Effective H/D =

1.04”, we placed the lead weight on top of the 60-mm fill height of the beads. It should

be noted that the particle size of the beads used in these experiments was 5 mm. This

34

was different from the other cases because the 2-mm particles were smaller than the hole

through which the lead weight was inserted onto the impeller shaft. Because of this, the

2-mm beads would escape upward through this hole and get on top of the lead weight,

making the lead weight sink into the particle bed. Therefore, in order to overcome this

problem, glass beads with a 5-mm particle size were used instead in the experiments in

this section.

All experiments in this part were performed using the standard impeller blades (Fig. 3a).

Additionally, all other process parameters were the same as those in the base case. Fig.

14 shows the time-averaged torque and the average adjusted power results for cases with

different fill heights of the glass beads as a function of the dimensionless shear rate. It

can be observed from Fig. 14a that the shallow bed (H/D = 0.30) of 5-mm beads exerts

the lowest normal load on the impeller blades, and thus the lowest values of torque are

measured. As the amount of material in the bladed mixer increases, the experimentally

measured torque generally increases with the fill height of the beads.

The glass beads filled up to the 85-mm fill height (H/D = 0.85) yields comparable results

to the case where the lead weight was placed on top of the material (Effective H/D = 0.85),

implying that the lead weight can be a good representative of the normal load applied on

the granular bed. In general, for the 104-mm fill height (Effective H/D = 1.04), the largest

torque values are measured due to the largest amount of hydrostatic pressure

experienced by the impeller blades during agitation. Similar trends for the average

35

adjusted power in the shallow bed granular system (H/D = 0.30) plotted as a function of

the dimensionless shear rate can be seen in Fig. 14b. A scale-up relationship for deep

granular beds varying the material fill height in a bladed mixer can be then developed

from the average torque and adjusted power information. From Eq. (2), it can be

rearranged to the following expression:

T⃗⃗ = 2π𝑅𝑐𝑦𝑙2 H⟨τ𝜃𝑟⟩ (9)

Note that the impeller torque (T⃗⃗ ) is proportional to the average shear stress τ𝜃𝑟

multiplied by the height (H) of the particle bed when the blades are rotating. In addition,

Remy et al. [5] found that, for non-cohesive materials, Coulomb’s law of friction was

observed and the shear stress was proportional to the normal stress. The normal stress is

in turn determined by the mass of powder in the mixer. In previous work, it has been

observed that the mass of powder in the mixer was a critical parameter in terms of the

powder flow patterns during the high shear mixing process [34]. In a bladed mixer, it was

observed that the average shear stress τ𝜃𝑟 could be related to the hydrostatic pressure

acting on the material [5]:

⟨𝜏𝜃𝑟⟩ = 𝜇𝜌bulk𝑔𝐻 (10)

where μ is the bulk friction coefficient, ρbulk is the bulk density of the granular bed, g is

the acceleration due to gravity, and H is the fill height of the bed. It will be assumed that

36

changes in the granular bed bulk density ρbulk during agitation are negligible, i.e., ρbulk is

approximately constant within the flow regimes we are investigating. This implies that

the height, H of the granular bed does not change with rotation rate. We indeed observed

in our experiments that when the blades are rotating the total height of the granular bed

is approximately the same as the granular bed initial height at a stationary state.

Substituting Eq. (10) into Eq. (9), the following relationship is derived subsequently:

T⃗⃗ = 2π𝜇𝜌bulk𝑔𝑅2cyl𝐻2 (11)

Expression (11) implies that the measured impeller torque scales by the square of the

material fill height T⃗⃗ 𝛼 𝐻2 in a bladed mixer when all other parameters are constant.

In Fig. 15, we normalize the experimentally measured time averaged torque T⃗⃗ and the

average adjusted power P* values in Fig. 14 by the square of the granular bed fill height

(H2) for each case. The normalized torque ⟨𝑇⟩/H2 results shown in Fig. 15a illustrate that

all of the normalized torque data points for the deep particle beds (H/D ≥ 0.60) essentially

collapse into one single curve. It is interesting to note that, even though the actual time-

averaged torque increases with an increase in the material fill height (see Fig. 14a), the

normalized torque ⟨𝑇⟩/𝐻2 for all deep granular bed cases (H/D ≥ 0.60) is lower than that

of the shallow bed case (H/D = 0.30). Moreover, the fact that the normalized torque

⟨𝑇⟩/𝐻2 values for the deep particle beds (H/D ≥ 0.60) collapse into one single curve, with

the exclusion of the shallow bed case (H/D = 0.30), implies that the impeller torque

exerted on deep granular beds behaves differently from that in a shallow bed. We

37

hypothesize this difference in the shallow bed case is because particles on the surface of

the shallow bed can form heaps and valleys freely when the impeller blades are rotating,

resulting in significant changes in the bed height and the bulk density especially at high

shear rates which can in turn affect the torque scaling relationship in Eq. (11). An

analogous principle can be applied to the impeller blade power consumption, and the

normalized adjusted power P*/H2 results are depicted in Fig. 15b. Similar trends are

observed for the normalized adjusted power P*/H2 both in the shallow and deep particle

bed cases for the entire range of the granular flow regime.

38

3.7 Figures for Chapter 3

Fig. 4. Normalized torque (�⃗� ∗) fluctuation profiles at steady state for different blade

rotational speeds: (a) 10 RPM, (b) 20 RPM, (c) 60 RPM, and (d) 200 RPM. The normalized

torque values shown in this figure are the instantaneous torques divided by the time-

averaged torque at each shear rate. Torque signal measurements were recorded every 0.125

s.

a)

b)

c)

d)

39

Fig. 5. Single-sided amplitude spectrums of the fast Fourier transform (FFT) torque signals

for different blade rotational speeds: (a) 10 RPM, (b) 20 RPM, (c) 60 RPM, and (d) 200

RPM. Torque signal measurements were recorded every 0.125 s.

0 1 2 3 40

0.002

0.004

0.006

0.008

0.01

0.012

Frequency, f (Hz)

|P(f

)|

0 1 2 3 40

0.002

0.004

0.006

0.008

0.01

0.012

Frequency, f (Hz)

|P(f

)|

0 1 2 3 40

0.002

0.004

0.006

0.008

0.01

0.012

Frequency, f (Hz)

|P(f

)|

0 1 2 3 40

0.002

0.004

0.006

0.008

0.01

0.012

Frequency, f (Hz)

|P(f

)|

b) a)

c) d)

40

Fig. 6. Granular flow properties as a function of the dimensionless shear rate (γ*) for “the

standard case”: (a) the time-averaged impeller torque and (b) the average adjusted power

(P*). The experiments were performed using red glass beads of 2-mm diameter loaded to a

30-mm fill height (shallow bed, H/D = 0.30) and the standard impeller blades (4 blades

with a 135° angle and 45 mm length). The adjusted power (P*) was obtained by subtracting

the power draw at a specific RPM for the empty mixer from the power draw for the impeller

in the mixer loaded with particles.

a)

b)

41

Fig. 7. Effect of the number of impeller blades on: (a) the average torque and (b) the

average adjusted power as a function of the dimensionless shear rate. Impeller blades used

in these sets of experiments have 135° angle and 45 mm length.

a)

b)

42

Fig. 8. Effect of the impeller blade angle on: (a) the average torque and (b) the average

adjusted power as a function of the dimensionless shear rate. Impeller blades used in these

sets of experiments have 4 blades and 45 mm length.

a)

b)

43

Fig. 9. Effect of the impeller blade length on: (a) the average torque and (b) the average

adjusted power as a function of the dimensionless shear rate. Impeller blades used in these

sets of experiments for both cases have 4 blades and 135° angle.

a)

b)

44

Fig. 10. Effect of the macroscopic friction coefficient (µ) of particles on: (a) the average

torque and (b) the average adjusted power as a function of the dimensionless shear rate.

Experiments were performed using the standard impeller blades and the glass beads with a

diameter of 2 mm loaded to a 30-mm fill height (H/D = 0.30).

a)

b)

45

Fig. 11. Effect of the impeller blade position in a deep granular bed (90-mm fill height) on:

(a) the time-averaged torque and (b) the average adjusted power as a function of the

dimensionless shear rate, compared to those values in a shallow bed (30-mm fill height just

covering the top tip of the blades). Blade position in a deep bed: top (H/D = 0.30), middle

(H/D = 0.60), and bottom (H/D = 0.90). Experiments were performed using the standard

impeller blades and the glass beads with a diameter of 2 mm.

a)

b)

46

Fig. 12. Effect of the particle size on: (a) the average torque and (b) the average adjusted

power drawn from experiments as a function of the shear rate γ°. Experiments were

performed using the standard impeller blades, and the granular bed was filled up to a 30-

mm height (H/D = 0.30) for all particle sizes.

a)

b)

47

Fig. 13. (a) The time-averaged normalized torque, ⟨𝑇⟩/𝑑, and (b) the average normalized

adjusted power, P*/d, plotted as a function of the shear rate γ°. Experiments were performed

using the standard impeller blades, and the granular bed was filled up to a 30-mm height

(H/D = 0.30) for all particle sizes. Note that the y-axis in Fig. 13b is multiplied by a factor

of 103.

a)

b)

48

Fig. 14. Effect of the amount of materials in a bladed mixer, reported as the H/D ratio, on:

(a) the time-averaged torque and (b) the average adjusted power as a function of the

dimensionless shear rate. Experiments were performed using the standard impeller blades

and the glass beads with a diameter of 5 mm. The effective H/D ratios indicate usage of a

lead weight instead of using actual particle mass.

a)

b)

49

Fig. 15. (a) The time-averaged torque ⟨�⃗� ⟩ and (b) the average adjusted power P* values in

Fig. 14 were normalized by square of the height (H2) of the granular material filled in a

bladed mixer plotted as a function of the dimensionless shear rate. Experiments were

performed using the standard impeller blades and the glass beads with a diameter of 5 mm.

The effective H/D ratios indicate usage of a lead weight instead of using actual particle

mass. Note that the y-axis in Fig. 15b is multiplied by a factor of 103.

a)

b)

50

4 CONCLUSION

Experiments were performed on monodisperse spherical glass beads flowing in a

cylindrical bladed mixer agitated by an impeller. Experimental measurements of the

torque exerted by the impeller blades on the granular bed were achieved by using a

torque table and a data acquisition system. Power consumed by the motor to move an

agitator through the particle bed was also experimentally measured using a power meter.

The effects of various impeller blade design configurations and material properties on the

agitation torque and power were examined as a function of the impeller blade rotational

speed. For “the base case”, which was composed of 2-mm diameter glass beads filled up

to a 30-mm height (H/D = 0.30) that just covered the span of the blades and considered

as a shallow bed using “the standard impeller blades”, the time-averaged torque values

at low shear rates were slightly increasing within the quasi-static regime of granular flow

behaviors. This was then followed by a linear increase in the torque in the intermediate

region as the shear rate increased. This trend for the impeller torque was similar to what

was observed for the adjusted power consumption. Data analysis in “the base case”

revealed that fluctuation of torque values was fairly constant in the quasi-static region

and increased with an increasing blade rotational speed in the intermediate flow regime.

FFT spectrums for “the base case” showed peaks at frequencies corresponding to the

impeller blade rotational rates.

51

This research demonstrates that the torque values and the power readings measured

from the mixing system were sensitive to several factors including the impeller blade

configurations (number, angle, and length of blades), the particle size and friction

coefficient, the blade position in a deep granular bed, and the fill height of the glass beads.

The time-averaged torque was dependent on the number of impeller blades used in the

mixer; the 2- and 4-bladed mixers exerted greater torque than the 1-bladed mixer on the

particle bed due to the larger blade-particle tangential forces. The torque values also

increased with an increase in the blade length and with a decrease in the blade pitch

angle. Furthermore, at higher fill heights of the granular material, greater torque values

and power readings could be measured. The highest torque and power measurements

were recorded for the case where the impeller blades were placed at the most bottom

position (H/D = 0.90) of the deep particle bed. This was because of the hydrostatic

pressure from the material on top of the blades. Moreover, the average torque and

adjusted power were a strong function of material properties including particle diameter

and surface roughness. Larger particle sizes and a greater friction coefficient resulted in

higher magnitudes of the torque and power. In addition, the time-averaged torque and

power consumption for the 2- and 5-mm bead cases qualitatively scaled with particle

diameter.

It was demonstrated that granular flow behaviors in a shallow bed were different from

those in the deep bed. As the amount of material in the bladed mixer increased, the

experimentally measured torque also increased. In the deep particle bed, the measured

52

torque was independent of the shear rate in a quasi-static regime and then linearly

increased as the blade rotational speed increased in the intermediate regime. Moreover,

the clear distinction between these two regions could be observed in the deep granular

bed, which could not be obviously differentiated in the shallow bed case. A scale-up

relationship for deep granular beds varying the material fill height in a bladed mixer was

then developed from the average torque and adjusted power information. The time

averaged torque and average adjusted power consumption qualitatively scaled with

square of the material fill height in a deep granular bed.

The results from research work in this chapter provide an insight on how the impeller

blade configurations and material properties affect the agitation torque and power

measured as a function of a process parameter in granular systems flowing in a bladed

mixer. These findings complement our preceding knowledge and provide better

understanding with regard to processing of solid particulate systems. At the same time

these results are for a limited number of parameters, and future work is needed to

validate that the results of this study hold for other parameter values. Further research

should be conducted to study the effect of other material properties (e.g. cohesive

particles), equipment configurations (e.g. different types of mixers), and other process

parameters on particulate flow and mixing in a bladed mixer. Experiments can be done

by adding and varying the amount of water in a granular bed to investigate the effect of

cohesion on measured torque and power consumed by the impeller blades. In addition,

results of the parametric sensitivity analysis for the impeller blade torque and power

53

consumption found in this research can be extended further by performing experiments

in larger scale mixers. DEM simulations in a similar geometry can also be carried out by

varying the aforementioned factors to provide information about torque and power.

54

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